03 - Std'07 - Mathematics.pdf - xi Contents Foreword iii Preface v Chapter 1 Integers 1 Chapter 2 Fractions and Decimals 29 Chapter 3 Data Handling 57

03 - Std'07 - Mathematics.pdf - xi Contents Foreword iii...

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Unformatted text preview: xi Contents Foreword iii Preface v Chapter 1 Integers 1 Chapter 2 Fractions and Decimals 29 Chapter 3 Data Handling 57 Chapter 4 Simple Equations 77 Chapter 5 Lines and Angles 93 Chapter 6 The Triangle and its Properties 113 Chapter 7 Congruence of Triangles 133 Chapter 8 Comparing Quantities 153 Chapter 9 Rational Numbers 173 Chapter 10 Practical Geometry 193 Chapter 11 Perimeter and Area 205 Chapter 12 Algebraic Expressions 229 Chapter 13 Exponents and Powers 249 Chapter 14 Symmetry 265 Chapter 15 Visualising Solid Shapes 277 Answers 293 Brain-Teasers 311 1.1 INTRODUCTION We have learnt about whole numbers and integers in Class VI. We know that integers form a bigger collection of numbers which contains whole numbers and negative numbers. What other differences do you find between whole numbers and integers? In this chapter, we will study more about integers, their properties and operations. First of all, we will review and revise what we have done about integers in our previous class. 1.2 RECALL We know how to represent integers on a number line. Some integers are marked on the number line given below. Can you write these marked integers in ascending order? The ascending order of these numbers is – 5, – 1, 3. Why did we choose – 5 as the smallest number? Some points are marked with integers on the following number line. Write these integers in descending order. The descending order of these integers is 14, 8, 3, ... The above number line has only a few integers filled. Write appropriate numbers at each dot. Chapter 1 Integers 2 MATHEMATICS TRY THESE 1. A number line representing integers is given below A B 2. C D E F –3 –2 G H I J K L M N O –3 and –2 are marked by E and F respectively. Which integers are marked by B, D, H, J, M and O? Arrange 7, –5, 4, 0 and – 4 in ascending order and then mark them on a number line to check your answer. We have done addition and subtraction of integers in our previous class. Read the following statements. On a number line when we (i) add a positive integer, we move to the right. (ii) add a negative integer, we move to the left. (iii) subtract a positive integer, we move to the left. (iv) subtract a negative integer, we move to the right. State whether the following statements are correct or incorrect. Correct those which are wrong: (i) When two positive integers are added we get a positive integer. (ii) When two negative integers are added we get a positive integer. (iii) When a positive integer and a negative integer are added, we always get a negative integer. (iv) Additive inverse of an integer 8 is (– 8) and additive inverse of (– 8) is 8. (v) For subtraction, we add the additive inverse of the integer that is being subtracted, to the other integer. (vi) (–10) + 3 = 10 – 3 (vii) 8 + (–7) – (– 4) = 8 + 7 – 4 Compare your answers with the answers given below: (i) Correct. For example: (a) 56 + 73 = 129 (b) 113 + 82 = 195 etc. Construct five more examples in support of this statement. (ii) Incorrect, since (– 6) + (– 7) = – 13, which is not a positive integer. The correct statement is: When two negative integers are added we get a negative integer. For example, (a) (– 56) + (– 73) = – 129 (b) (– 113) + (– 82) = – 195, etc. Construct five more examples on your own to verify this statement. INTEGERS (iii) Incorrect, since – 9 + 16 = 7, which is not a negative integer. The correct statement is : When one positive and one negative integers are added, we take their difference and place the sign of the bigger integer. The bigger integer is decided by ignoring the signs of both the integers. For example: (a) (– 56) + (73) = 17 (b) (– 113) + 82 = – 31 (c) 16 + (– 23) = – 7 (d) 125 + (– 101) = 24 Construct five more examples for verifying this statement. (iv) Correct. Some other examples of additive inverse are as given below: Integer 10 –10 76 –76 Additive inverse –10 10 –76 76 Thus, the additive inverse of any integer a is – a and additive inverse of (– a) is a. (v) Correct. Subtraction is opposite of addition and therefore, we add the additive inverse of the integer that is being subtracted, to the other integer. For example: (a) 56 – 73 = 56 + additive inverse of 73 = 56 + (–73) = –17 (b) 56 – (–73) = 56 + additive inverse of (–73) = 56 + 73 = 129 (c) (–79) – 45 = (–79) + (– 45) = –124 (d) (–100) – (–172) = –100 + 172 = 72 etc. Write atleast five such examples to verify this statement. Thus, we find that for any two integers a and b, a – b = a + additive inverse of b = a + (– b) and a – (– b) = a + additive inverse of (– b) = a + b (vi) Incorrect, since (–10) + 3 = –7 and 10 – 3 = 7 therefore, (–10) + 3 ≠ 10 – 3 (vii) Incorrect, since, 8 + (–7) – (– 4) = 8 + (–7) + 4 = 1 + 4 = 5 and 8 + 7 – 4 = 15 – 4 = 11 However, 8 + (–7) – (– 4) = 8 – 7 + 4 TRY THESE We have done various patterns with numbers in our previous class. Can you find a pattern for each of the following? If yes, complete them: (a) 7, 3, – 1, – 5, _____, _____, _____. (b) – 2, – 4, – 6, – 8, _____, _____, _____. (c) 15, 10, 5, 0, _____, _____, _____. (d) – 11, – 8, – 5, – 2, _____, _____, _____. Make some more such patterns and ask your friends to complete them. 3 4 MATHEMATICS EXERCISE 1.1 1. Following number line shows the temperature in degree celsius (°C) at different places on a particular day. Lahulspiti –10 Shimla Srinagar –5 0 Ooty 5 10 Bangalore 15 20 25 (a) Observe this number line and write the temperature of the places marked on it. (b) What is the temperature difference between the hottest and the coldest places among the above? (c) What is the temperature difference between Lahulspiti and Srinagar? (d) Can we say temperature of Srinagar and Shimla taken together is less than the temperature at Shimla? Is it also less than the temperature at Srinagar? 2. In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jack’s scores in five successive rounds were 25, – 5, – 10, 15 and 10, what was his total at the end? 3. At Srinagar temperature was – 5°C on Monday and then it dropped by 2°C on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday, it rose by 4°C. What was the temperature on this day? 4. A plane is flying at the height of 5000 m above the sea level. At a particular point, it is exactly above a submarine floating 1200 m below the sea level. What is the vertical distance between them? 5. Mohan deposits Rs 2,000 in his bank account and withdraws Rs 1,642 from it, the next day. If withdrawal of amount from the account is represented by a negative integer, then how will you represent the amount deposited? Find the balance in Mohan’s account after the withdrawal. 6. Rita goes 20 km towards east from a point A to the point B. From B, she moves 30 km towards west along the same road. If the distance towards east is represented by a positive integer then, how will you represent the distance travelled towards west? By which integer will you represent her final position from A? INTEGERS 7. In a magic square each row, column and diagonal have the same sum. Check which of the following is a magic square. 5 –1 –4 1 –10 –5 –2 7 –4 –3 –2 0 3 –3 –6 4 –7 (i) 0 (ii) 8. Verify a – (– b) = a + b for the following values of a and b. (i) a = 21, b = 18 (iii) a = 75, b = 84 (ii) a = 118, b = 125 (iv) a = 28, b = 11 9. Use the sign of >, < or = in the box to make the statements true. (a) (– 8) + (– 4) (–8) – (– 4) (b) (– 3) + 7 – (19) 15 – 8 + (– 9) (c) 23 – 41 + 11 23 – 41 – 11 (d) 39 + (– 24) – (15) 36 + (– 52) – (– 36) (e) – 231 + 79 + 51 –399 + 159 + 81 10. A water tank has steps inside it. A monkey is sitting on the topmost step (i.e., the first step). The water level is at the ninth step. (i) He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will he reach the water level? (ii) After drinking water, he wants to go back. For this, he jumps 4 steps up and then jumps back 2 steps down in every move. In how many jumps will he reach back the top step? (iii) If the number of steps moved down is represented by negative integers and the number of steps moved up by positive integers, represent his moves in part (i) and (ii) by completing the following; (a) – 3 + 2 – ... = – 8 (b) 4 – 2 + ... = 8. In (a) the sum (– 8) represents going down by eight steps. So, what will the sum 8 in (b) represent? 1.3 PROPERTIES OF ADDITION AND SUBTRACTION OF INTEGERS 1.3.1 Closure under Addition We have learnt that sum of two whole numbers is again a whole number. For example, 17 + 24 = 41 which is again a whole number. We know that, this property is known as the closure property for addition of the whole numbers. 5 6 MATHEMATICS Let us see whether this property is true for integers or not. Following are some pairs of integers. Observe the following table and complete it. Statement (i) (ii) (iii) (iv) (v) (vi) (vii) 17 + 23 = 40 (–10) + 3 = _____ (– 75) + 18 = _____ 19 + (– 25) = – 6 27 + (– 27) = _____ (– 20) + 0 = _____ (– 35) + (– 10) = _____ Observation Result is an integer ______________ ______________ Result is an integer ______________ ______________ ______________ What do you observe? Is the sum of two integers always an integer? Did you find a pair of integers whose sum is not an integer? Since addition of integers gives integers, we say integers are closed under addition. In general, for any two integers a and b, a + b is an integer. 1.3.2 Closure under Subtraction What happens when we subtract an integer from another integer? Can we say that their difference is also an integer? Observe the following table and complete it: Statement Observation (i) 7 – 9 = – 2 Result is an integer (ii) 17 – (– 21) = _______ ______________ (iii) (– 8) – (–14) = 6 Result is an integer (iv) (– 21) – (– 10) = _______ ______________ (v) 32 – (–17) = _______ ______________ (vi) (– 18) – (– 18) = _______ ______________ (vii) (– 29) – 0 = _______ ______________ What do you observe? Is there any pair of integers whose difference is not an integer? Can we say integers are closed under subtraction? Yes, we can see that integers are closed under subtraction. Thus, if a and b are two integers then a – b is also an intger. Do the whole numbers satisfy this property? INTEGERS 1.3.3 Commutative Property We know that 3 + 5 = 5 + 3 = 8, that is, the whole numbers can be added in any order. In other words, addition is commutative for whole numbers. Can we say the same for integers also? We have 5 + (– 6) = –1 and (– 6) + 5 = –1 So, 5 + (– 6) = (– 6) + 5 Are the following equal? (i) (– 8) + (– 9) and (– 9) + (– 8) (ii) (– 23) + 32 and 32 + (– 23) (iii) (– 45) + 0 and 0 + (– 45) Try this with five other pairs of integers. Do you find any pair of integers for which the sums are different when the order is changed? Certainly not. Thus, we conclude that addition is commutative for integers. In general, for any two integers a and b, we can say a+b=b+a  We know that subtraction is not commutative for whole numbers. Is it commutative for integers? Consider the integers 5 and (–3). Is 5 – (–3) the same as (–3) –5? No, because 5 – ( –3) = 5 + 3 = 8, and (–3) – 5 = – 3 – 5 = – 8. Take atleast five different pairs of integers and check this. We conclude that subtraction is not commutative for integers. 1.3.4 Associative Property Observe the following examples: Consider the integers –3, –2 and –5. Look at (–5) + [(–3) + (–2)] and [(–5) + (–3)] + (–2). In the first sum (–3) and (–2) are grouped together and in the second (–5) and (–3) are grouped together. We will check whether we get different results. (–5) + [(–3) + (–2)] [(–5) + (–3)] + (–2) 7 8 MATHEMATICS In both the cases, we get –10. i.e., (–5) + [(–3) + (–2)] = [(–5) + (–2)] + (–3) Similarly consider –3 , 1 and –7. ( –3) + [1 + (–7)] = –3 + __________ = __________ [(–3) + 1] + (–7) = –2 + __________ = __________ Is (–3) + [1 + (–7)] same as [(–3) + 1] + (–7)? Take five more such examples. You will not find any example for which the sums are different. This shows that addition is associative for integers. In general for any integers a, b and c, we can say a + (b + c) = (a + b) + c 1.3.5 Additive Identity When we add zero to any whole number, we get the same whole number. Zero is an additive identity for whole numbers. Is it an additive identity again for integers also? Observe the following and fill in the blanks: (i) (– 8) + 0 = – 8 (ii) 0 + (– 8) = – 8 (iii) (–23) + 0 = _____ (iv) 0 + (–37) = –37 (v) 0 + (–59) = _____ (vi) 0 + _____ = – 43 (vii) – 61 + _____ = – 61 (viii) _____ + 0 = _____ The above examples show that zero is an additive identity for integers. You can verify it by adding zero to any other five integers. In general, for any integer a a+0=a=0+a TRY THESE 1. Write a pair of integers whose sum gives (a) a negative integer (b) zero (c) an integer smaller than both the integers. (e) an integer greater than both the integers. (d) an integer smaller than only one of the integers. 2. Write a pair of integers whose difference gives (a) a negative integer. (c) an integer smaller than both the integers. (e) an integer greater than both the integers. (b) zero. (d) an integer greater than only one of the integers. INTEGERS EXAMPLE 1 Write down a pair of integers whose SOLUTION (a) sum is –3 (b) difference is –5 (c) difference is 2 (d) sum is 0 (a) (–1) + (–2) = –3 or (–5) + 2 = –3 (b) (–9) – (– 4) = –5 or (–2) – 3 = –5 (c) (–7) – (–9) = 2 or 1 – (–1) = 2 (d) (–10) + 10 = 0 or 5 + (–5) = 0 Can you write more pairs in these examples? EXERCISE 1.2 1. Write down a pair of integers whose: (a) sum is –7 (b) difference is –10 (c) sum is 0 2. (a) Write a pair of negative integers whose difference gives 8. (b) Write a negative integer and a positive integer whose sum is –5. (c) Write a negative integer and a positive integer whose difference is –3. 3. In a quiz, team A scored – 40, 10, 0 and team B scored 10, 0, – 40 in three successive rounds. Which team scored more? Can we say that we can add integers in any order? 4. Fill in the blanks to make the following statements true: (i) (–5) + (............) = (– 8) + (............) (ii) –53 + ............ = –53 (iii) 17 + ............ = 0 (iv) [13 + (– 12)] + (............) = ............ + [(–12) + (–7)] (v) (– 4) + [............ + (–3)] = [............ + 15] + ............ 1.4 MULTIPLICATION OF INTEGERS We can add and subtract integers. Let us now learn how to multiply integers. 1.4.1 Multiplication of a Positive and a Negative Integer We know that multiplication of whole numbers is repeated addition. For example, 5 + 5 + 5 = 3 × 5 = 15 Can you represent addition of integers in the same way? 9 10 MATHEMATICS TRY THESE We have from the following number line, (–5) + (–5) + (–5) = –15 Find: 4 × (– 8), 8 × (–2), 3 × (–7), 10 × (–1) using number line. –20 –15 0 (–5) + (–5) + (–5) = 3 × (–5) 3 × (–5) = –15 (– 4) + (– 4) + (– 4) + (– 4) + (– 4) = 5 × (– 4) = –20 –16 –20 And –5 But we can also write Therefore, Similarly –10 –12 –8 –4 0 (–3) + (–3) + (–3) + (–3) = __________ = __________ Also, (–7) + (–7) + (–7) = __________ = __________ Let us see how to find the product of a positive integer and a negative integer without using number line. Let us find 3 × (–5) in a different way. First find 3 × 5 and then put minus sign (–) before the product obtained. You get –15. That is we find – (3 × 5) to get –15. Similarly, 5 × (– 4) = – (5×4) = – 20. Find in a similar way, 4 × (– 8) = _____ = _____ 3 × (– 7) = _____ = _____ 6 × (– 5) = _____ = _____ 2 × (– 9) = _____ = _____ Using this method we thus have, TRY THESE Find: (i) 6 × (–19) (ii) 12 × (–32) (iii) 7 × (–22) We have, 10 × (– 43) = _____ – (10 × 43) = – 430 Till now we multiplied integers as (positive integer) × (negative integer). Let us now multiply them as (negative integer) × (positive integer). We first find –3 × 5. To find this, observe the following pattern: 3 × 5 = 15 2 × 5 = 10 = 15 – 5 1 × 5 = 5 = 10 – 5 0×5=0=5–5 So, –1 × 5 = 0 – 5 = –5 INTEGERS –2 × 5 = –5 – 5 = –10 –3 × 5 = –10 – 5 = –15 We already have 3 × (–5) = –15 So we get (–3) × 5 = –15 = 3 × (–5) Using such patterns, we also get (–5) × 4 = –20 = 5 × (– 4) Using patterns, find (– 4) × 8, (–3) × 7, (– 6) × 5 and (– 2) × 9 Check whether, (– 4) × 8 = 4 × (– 8), (– 3) × 7 = 3 × (–7), (– 6) × 5 = 6 × (– 5) and (– 2) × 9 = 2 × (– 9) Using this we get, (–33) × 5 = 33 × (–5) = –165 We thus find that while multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (–) before the product. We thus get a negative integer. TRY THESE 1. Find: (a) 15 × (–16) (b) 21 × (–32) (c) (– 42) × 12 (d) –55 × 15 (b) (–23) × 20 = 23 × (–20) 2. Check if (a) 25 × (–21) = (–25) × 21 Write five more such examples. In general, for any two positive integers a and b we can say a × (– b) = (– a) × b = – (a × b) 1.4.2 Multiplication of two Negative Integers Can you find the product (–3) × (–2)? Observe the following: –3 × 4 = – 12 –3 × 3 = –9 = –12 – (–3) –3 × 2 = – 6 = –9 – (–3) –3 × 1 = –3 = – 6 – (–3) –3 × 0 = 0 = –3 – (–3) –3 × –1 = 0 – (–3) = 0 + 3 = 3 –3 × –2 = 3 – (–3) = 3 + 3 = 6 Do you see any pattern? Observe how the products change. 11 12 MATHEMATICS Based on this observation, complete the following: –3 × –3 = _____ –3 × – 4 = _____ Now observe these products and fill in the blanks: – 4 × 4 = –16 TRY THESE – 4 × 3 = –12 = –16 + 4 (i) Starting from (–5) × 4, find (–5) × (– 6) (ii) Starting from (– 6) × 3, find (– 6) × (–7) – 4 × 2 = _____ = –12 + 4 – 4 × 1 = _____ – 4 × 0 = _____ – 4 × (–1) = _____ – 4 × (–2) = _____ – 4 × (–3) = _____ From these patterns we observe that, (–3) × (–1) = 3 = 3 × 1 (–3) × (–2) = 6 = 3 × 2 (–3) × (–3) = 9 = 3 × 3 and (– 4) × (–1) = 4 = 4 × 1 So, (– 4) × (–2) = 4 × 2 = _____ (– 4) × (–3) = _____ = _____ So observing these products we can say that the product of two negative integers is a positive integer. We multiply the two negative integers as whole numbers and put the positive sign before the product. Thus, we have (–10) × (–12) = 120 Similarly (–15) × (– 6) = 90 In general, for any two positive integers a and b, (– a) × (– b) = a × b TRY THESE Find: (–31) × (–100), (–25) × (–72), (–83) × (–28) Game 1 (i) Take a board marked from –104 to 104 as shown in the figure. (ii) Take a bag containing two blue and two red dice. Number of dots on the blue dice indicate positive integers and number of dots on the red dice indicate negative integers. (iii) Every player will place his/her counter at zero. (iv) Each player will take out two dice at a time from the bag and throw them. INTEGERS 104 83 82 61 60 39 38 17 16 –5 –6 –27 –28 – 49 – 50 – 71 –72 – 93 – 94 103 84 81 62 59 40 37 18 15 –4 –7 –26 –29 – 48 –51 – 70 –73 – 92 – 95 102 85 80 63 58 41 36 19 14 –3 –8 –25 –30 – 47 –52 – 69 –74 – 91 – 96 101 86 79 64 57 42 35 20 13 –2 –9 –24 –31 – 46 –53 – 68 –75 – 90 – 97 100 87 78 65 56 43 34 21 12 –1 –10 –23 –32 – 45 –54 – 67 –76 – 89 – 98 99 98 88 89 77 76 66 67 55 54 44 45 33 32 22 23 11 10 0 1 –11 –12 –22 –21 –33 –34 – 44 – 43 –55 –56 – 66 – 65 –77 –78 – 88 – 87 – 99 – 100 97 90 75 68 53 46 31 24 9 2 –13 –20 –35 – 42 –57 – 64 –79 – 86 –101 96 91 74 69 52 47 30 25 8 3 –14 –19 –36 – 41 –58 – 63 –80 – 85 –102 95 92 73 70 51 48 29 26 7 4 –15 –18 –37 – 40 –59 – 62 –81 – 84 –103 94 93 72 71 50 49 28 27 6 5 –16 –17 –38 –39 – 60 – 61 –82 – 83 –104 (v) After every throw, the player has to multiply the numbers marked on the dice. (vi) If the product is a positive integer then the player will move his counter towards 104; if the product is a negative integer then the player will move his counter towards –104. (vii) The player who reaches 104 first is the winner. 13 14 MATHEMATICS 1.4.3 Product of three or more Negative Integers Euler in his book Ankitung zur We observed that the product of two negative integers is a positive integer. Algebra(1770), was one of What will be the product of three negative integers? Four negative integers? the first mathematicians to Let us observe the following examples: attempt to prove (a) (– 4) × (–3) = 12 (–1) × (...
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